Question: Determine how many solutions exist for the system of equations. ${3x+3y = -9}$ ${-6x+3y = 27}$
Convert both equations to slope-intercept form: ${3x+3y = -9}$ $3x{-3x} + 3y = -9{-3x}$ $3y = -9-3x$ $y = -3-x$ ${y = -x-3}$ ${-6x+3y = 27}$ $-6x{+6x} + 3y = 27{+6x}$ $3y = 27+6x$ $y = 9+2x$ ${y = 2x+9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -x-3}$ ${y = 2x+9}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.